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Simplifying 7k2 + 54k + 13 = 5 Reorder the terms: 13 + 54k + 7k2 = 5 Solving 13 + 54k + 7k2 = 5 Solving for variable 'k'. Reorder the terms: 13 + -5 + 54k + 7k2 = 5 + -5 Combine like terms: 13 + -5 = 8 8 + 54k + 7k2 = 5 + -5 Combine like terms: 5 + -5 = 0 8 + 54k + 7k2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 1.142857143 + 7.714285714k + k2 = 0 Move the constant term to the right: Add '-1.142857143' to each side of the equation. 1.142857143 + 7.714285714k + -1.142857143 + k2 = 0 + -1.142857143 Reorder the terms: 1.142857143 + -1.142857143 + 7.714285714k + k2 = 0 + -1.142857143 Combine like terms: 1.142857143 + -1.142857143 = 0.000000000 0.000000000 + 7.714285714k + k2 = 0 + -1.142857143 7.714285714k + k2 = 0 + -1.142857143 Combine like terms: 0 + -1.142857143 = -1.142857143 7.714285714k + k2 = -1.142857143 The k term is 7.714285714k. Take half its coefficient (3.857142857). Square it (14.87755102) and add it to both sides. Add '14.87755102' to each side of the equation. 7.714285714k + 14.87755102 + k2 = -1.142857143 + 14.87755102 Reorder the terms: 14.87755102 + 7.714285714k + k2 = -1.142857143 + 14.87755102 Combine like terms: -1.142857143 + 14.87755102 = 13.734693877 14.87755102 + 7.714285714k + k2 = 13.734693877 Factor a perfect square on the left side: (k + 3.857142857)(k + 3.857142857) = 13.734693877 Calculate the square root of the right side: 3.706034792 Break this problem into two subproblems by setting (k + 3.857142857) equal to 3.706034792 and -3.706034792.Subproblem 1
k + 3.857142857 = 3.706034792 Simplifying k + 3.857142857 = 3.706034792 Reorder the terms: 3.857142857 + k = 3.706034792 Solving 3.857142857 + k = 3.706034792 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3.857142857' to each side of the equation. 3.857142857 + -3.857142857 + k = 3.706034792 + -3.857142857 Combine like terms: 3.857142857 + -3.857142857 = 0.000000000 0.000000000 + k = 3.706034792 + -3.857142857 k = 3.706034792 + -3.857142857 Combine like terms: 3.706034792 + -3.857142857 = -0.151108065 k = -0.151108065 Simplifying k = -0.151108065Subproblem 2
k + 3.857142857 = -3.706034792 Simplifying k + 3.857142857 = -3.706034792 Reorder the terms: 3.857142857 + k = -3.706034792 Solving 3.857142857 + k = -3.706034792 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3.857142857' to each side of the equation. 3.857142857 + -3.857142857 + k = -3.706034792 + -3.857142857 Combine like terms: 3.857142857 + -3.857142857 = 0.000000000 0.000000000 + k = -3.706034792 + -3.857142857 k = -3.706034792 + -3.857142857 Combine like terms: -3.706034792 + -3.857142857 = -7.563177649 k = -7.563177649 Simplifying k = -7.563177649Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.151108065, -7.563177649}
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